On the relative category in the brake orbits problem
DOI:
https://doi.org/10.12775/TMNA.2022.057Keywords
Lusternik-Schnirelmann category, variational inequalities, brake orbitsAbstract
In this paper %dedicated to the memory of Edward Fadell and Sufian Husseini we show how the notion of the Lusternik-Schnirelmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the $N$-dimensional unit disk. The estimate of the relative category of the set of chords with endpoints on the $(N-1)$-unit sphere was shown to the third author by Fadell and Husseini while he was visiting the University of Wisconsin at Madison.References
W. Bos, Kritische Sehenen auf Riemannischen Elementarraumstücken, Math. Ann. 151 (1963), 431–451.
D. Corona, A multiplicity result for orthogonal geodesic chords in Finsler disks, Discrete Contin. Dyn. Syst. 41 (2021), no. 11, 5329–5357
D. Corona, A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions, J. Fixed Point Theory Appl. 22 (2020), 60.
D. Corona and F. Giannoni, Brake orbits for Hamiltonian systems of classical type via geodesics in singular Finsler metrics, Adv. Nonlinear Anal. 11 (2022), 1223–1248.
D. Corona and R. Giambò, Multiple brake-orbits for Hamiltonian systems of classical type, (in preparation).
E. Fadell and S. Husseini, Relative cohomological index theories, Adv. Math. 64 (1987), 1–31.
G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 259–281.
R. Giambò, F. Giannoni and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds, Adv. Differential Equations 10 (2005), 931–960.
R. Giambò, F. Giannoni and P. Piccione, Multiple Brake orbits and homoclinics in Riemannian manifolds, Arch. Ration. Mech. Anal. 200 (2011), 691–724.
R. Giambò, F. Giannoni, and P. Piccione, Examples with minimal number of brake orbits and homoclinics in annular potential regions, J. Differential Equations 256 (2014), 2677–2690.
R. Giambò, F. Giannoni and P. Piccione, Morse Theory for geodesics in singular con[12] R. Giambò, F. Giannoni and P. Piccione, Multiple brake orbits in m-dimensional disks, Calc. Var. Partial Differential Equations 54 (2015), 2553–2580.
R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords and a proof of Seifert’s conjecture on brake orbits, preprint 2021, arXiv:2002.09687.
R. Giambò, F. Giannoni, and P. Piccione, Multiple ortoghonal geodesics chords in nonconvex Riemannian disks using obstacles, Calc. Var. Partial Differential Equations 57 (2018), 117.
F. Giannoni, Multiplicity of principal bounce trajectories with prescribed minimal period on Riemannian manifolds, Differential Integral Equations 6 (1993), 1451–1480.
N.H. Kuiper, Double normal of convex values, Israel J. Math. 2 (1964), 71–80.
H. Liu and Y. Long, Resonance identity for symmetric closed characteristics on symmetric convex Hamiltonian energy hypersurfaces and its applications, J. Differential Equations 255 (2013), 2952–2980.
C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math. 67 (2014), 1563–1604.
C.Liu, Index Theory in Nonlinear Analysis, Springer, Berlin, 2019.
Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domain, Adv. Math. 203 (2006), no. 2, 568–635.
Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in R2n Adv. Math. 155 (2002), no. 2, 317–368.
L. Lusternik and L. Schnirelman, Methodes Topologiques dans les Problemes Variationelles, Hermann, 1934.
A. Marino and D. Scolozzi, Geodetiche con ostacolo, Boll. Unione Mat. Ital. 6 (1983), no. 2-B, 1–31.
R. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132.
P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202.
P.H. Rabinowitz, Critical point theory and applications to differential equations: a survey, Topological Nonlinear Analysis, vol. 15, (M. Matzeu and A. Vignoli, eds.), Birkhäuser, Boston, 1995, pp. 464–513.
H. Seifert, Periodische Bewegungen Machanischer Systeme, Math. Z. 51 (1948), 197–216.
E. H. Spanier, Algebraic Topology, Springer–Verlag, New York, 1989.
A. Weinstein, Periodic orbits for convex hamiltonian systems, Ann. of Math. (2) 108 (1978), no. 3, 507–18.
D. Zhang, Brake type closed characteristics on reversible compact convex hypersurfaces in R2n , Nonlinear Anal. 74 (2011), 3149-3158.
D. Zhang and C.Liu, Multiplicity of brake orbits on compact convex symmetric reversible hypersurfaces in R2n for n ≥ 4, Proc. Lond. Math. Soc. (3) 107 (2013), 1-38
D. Zhang and C.Liu, Multiple brake orbits on compact convex symmetric reversible hypersurfaces in R2n , Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 3, 531–554.
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Copyright (c) 2023 Dario Corona, Roberto Giambò, Fabio Giannoni, Paolo Piccione
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